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Theorem dedth3h 4132
Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4131. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
dedth3h.1 (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃𝜏))
dedth3h.2 (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏𝜂))
dedth3h.3 (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂𝜁))
dedth3h.4 𝜁
Assertion
Ref Expression
dedth3h ((𝜑𝜓𝜒) → 𝜃)

Proof of Theorem dedth3h
StepHypRef Expression
1 dedth3h.1 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃𝜏))
21imbi2d 330 . . 3 (𝐴 = if(𝜑, 𝐴, 𝐷) → (((𝜓𝜒) → 𝜃) ↔ ((𝜓𝜒) → 𝜏)))
3 dedth3h.2 . . . 4 (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏𝜂))
4 dedth3h.3 . . . 4 (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂𝜁))
5 dedth3h.4 . . . 4 𝜁
63, 4, 5dedth2h 4131 . . 3 ((𝜓𝜒) → 𝜏)
72, 6dedth 4130 . 2 (𝜑 → ((𝜓𝜒) → 𝜃))
873impib 1260 1 ((𝜑𝜓𝜒) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  ifcif 4077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-if 4078
This theorem is referenced by:  dedth3v  4135  faclbnd4lem2  13064  dvdsle  15013  gcdaddm  15227  ipdiri  27655  hvaddcan  27897  hvsubadd  27904  norm3dif  27977  omlsii  28232  chjass  28362  ledi  28369  spansncv  28482  pjcjt2  28521  pjopyth  28549  hoaddass  28611  hocsubdir  28614  hoddi  28819
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